A256553 - OEIS

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A256553

Triangle T(n,k) in which the n-th row contains the increasing list of distinct orders of degree-n permutations; n>=0, 1<=k<=A009490(n).

1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 10, 12, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20, 21, 30

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Alois P. Heinz, Rows n = 0..60, flattened

FORMULA

Sum_{k>=0} T(n,k)*A256554(n,k) = A181844(n).

T(n,k) = k for n>0 and 1<=k<=n.

EXAMPLE

Triangle T(n,k) begins:

1, 2;

1, 2, 3;

1, 2, 3, 4;

1, 2, 3, 4, 5, 6;

1, 2, 3, 4, 5, 6, 7, 10, 12;

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15;

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20;

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20, 21, 30;

MAPLE

b:= proc(n, i) option remember; `if`(n=0 or i=1, x,

b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),

t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))

end:

T:= n->(p->seq((h->`if`(h=0, [][], i))(coeff(p, x, i))

, i=1..degree(p)))(b(n$2)):

seq(T(n), n=0..12);

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x,

b[n, i - 1] + Function[p, Sum[Coefficient[p, x, t]*x^LCM[t, i],

{t, 1, Exponent[p, x]}]][Sum[b[n - i*j, i - 1], {j, 1, n/i}]]];

T[n_] := Function[p, Table[Function[h, If[h == 0, Nothing, i]][

Coefficient[p, x, i]], {i, 1, Exponent[p, x]}]][b[n, n]];

Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jul 15 2021, after Alois P. Heinz *)

CROSSREFS

Row sums give A060179.

Row lengths give A009490.

Last elements of rows give A000793.

Main diagonal gives A000027.

Cf. A181844, A256067, A256554.

Sequence in context: A066041 A194965 A243712 * A194896 A212721 A222417

Adjacent sequences: A256550 A256551 A256552 * A256554 A256555 A256556

KEYWORD

nonn,look,tabf

AUTHOR

Alois P. Heinz, Apr 01 2015

STATUS

approved